IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 62, NO. 5, MAY 2013 1259
A Low-Cost Instrument for the AccurateMeasurement of Resonances in Microwave Cavities
Simone Corbellini and Roberto M. Gavioso
Abstract—Since vector network analysis became available, itdemonstrated its accuracy and versatility in a variety of appli-cations, including the accurate measurement of resonances inmicrowave cavities. Unfortunately, the high cost and bulkinessof vector network analyzers (VNAs) set a limit to their appli-cability. This paper presents and discusses the design and theinitial performance tests of a simplified instrument which mayrepresent a valid alternative to VNAs in those applications wherethe high quality factors of a microwave resonator have to be de-termined with comparable accuracy but at low cost, allowing fieldportability and the embedment in a more complex measurementsystem. Triggered by the recent development of quasi-sphericalmicrowave resonators and their successful utilization in gasmetrology, we choose the extremely precise measurement of theireigenfrequencies as a suitable test bench to validate the specifica-tions and assess the performance of the proposed instrument.
Index Terms—Dielectric constant, microwave hygrometry,microwave resonances, permittivity, quasi-spherical resonators(QSRs).
I. INTRODUCTION
THE USE of microwave resonators for fundamental metrol-ogy dates back to the accurate determination of the speed
of light in vacuum by Essen in 1948 [1] using a cylindricalcavity. Also, accurate measurements of the permittivity of di-electric materials and standards, based on the use of microwaveresonant cavities of cylindrical geometry, are well established[2]. Recently, sapphire whispering gallery thermometers havebeen proposed as potential replacement of standard platinumresistance thermometers in industrial applications [3]. Finally,the significant development of both the theory and the exper-imental practice of quasi-spherical resonators (QSRs) [4], [5]has led to several achievements which include the following:the development of innovative primary standards for pressure[6] and temperature [7]; a relevant reduction of the uncertaintyassociated to the Boltzmann constant k, favoring a new defi-nition of the kelvin [8]; and the accurate measurement of thedielectric constant of pure gases and mixtures, with promising
Manuscript received June 22, 2012; revised September 15, 2012; acceptedOctober 29, 2012. Date of publication March 7, 2013; date of current versionApril 3, 2013. This work was supported by European Association of NationalMetrology Institutes and the European Union, under the European MetrologyResearch Programme. The Associate Editor coordinating the review process forthis paper was Dr. Wendy Van Moer.
S. Corbellini is with the Dipartimento di Elettronica e Telecomuni-cazioni, Politecnico di Torino, 10129 Torino, Italy (e-mail: [email protected]).
R. M. Gavioso is with the Thermodynamics Division, Istituto Nazionale diRicerca Metrologica, 10135 Torino, Italy (e-mail: [email protected]).
Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TIM.2013.2245038
applicability to extend the temperature and pressure workingrange of hygrometers and humidity standards [9], [10].
To be effective, all these applications need an extremely pre-cise and accurate determination of the frequency and half-widthof the electromagnetic modes excited and detected within aresonant microwave cavity and/or a dielectric sample. In recentacoustic/microwave determinations of k [11]–[13], the meanradius of an ellipsoidal resonator, typically on the order of a fewcentimeters, was estimated with an overall uncertainty as low as10 nm [14]. This achievement, in addition to a suitably refinedmodel of the major geometrical and waveguide perturbationsof the cavity field, required the capability to determine theresonance frequencies well below the part-per-million (ppm)level of accuracy.
In other metrological applications, like acoustic gas ther-mometry and the determination of the dielectric permittivity ofpure gases and mixtures, the measurement of squared frequencyratios, rather than absolute frequencies, relaxes the accuracyrequirements, as the ratioing operation reduces the uncertaintycontribution of a number of possible systematic error sources.Even in this case, part-per-billion (ppb) precision in the deter-mination of microwave eigenfrequencies is required to preservethe overall sensitivity of the method.
Vector network analyzers (VNAs) [15] are the electronic in-struments used to accomplish these rather demanding tasks, asthey can easily achieve a relative accuracy in the measurementof resonance frequencies of a few parts in 109. However, themajority of current state-of-the-art VNAs can be bulky instru-ments not preferably used in field conditions. In recent years,new portable VNAs have started appearing on the market;however, even these VNAs have unnecessary capability for thisspecific type of measurement, and their cost remains consider-ably high, thus limiting the development and the deployment ofpractical applications of microwave resonators. As an exampleof one such application, the simple and rugged design of QSRsmakes them potentially useful as microwave hygrometers forthe determination of high mixing ratios in moist hydrocarbonmixtures at high temperatures and pressures [16].
To achieve the accurate frequency measurements required bythis application, many of the VNA features are overspecified,redundant, and unnecessary, while a low-cost (LC) portableinstrument is needed. However, while these tasks are readilyaccomplishable by using a VNA, they become very ambitiousand challenging for an LC instrument.
In the remainder of this work, we describe and discuss thedevelopment and the performance of a simplified microwaveinstrument which may compete with VNAs. As a basic re-quired specification of this instrument, we fix the target relative
1260 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 62, NO. 5, MAY 2013
uncertainty in the determination of the frequency of QSR reso-nances at the level of 50 ppb. The corresponding relative con-tribution to the uncertainty of the squared microwave frequencyratios used to determine the thermal expansion of the internaldimensions of a primary acoustic thermometer, or the dielectricconstant of a gas, would thus be limited below 150 ppb. Inabsolute terms, such performance would be equivalent to thefollowing: an uncertainty contribution between 50 and 120 μKfor thermodynamic temperatures determined between 300 and600 K and an uncertainty contribution between 0.03 and 0.3 Kfor the frost point temperatures determined by measuringthe permittivity of a humid nitrogen mixture between 270 and240 K [9]. Additional required specifications with regard tothe operating frequency range, frequency resolution, dynamicrange, and noise background are discussed hereinafter. Finally,we keep the requirement of limiting the power consumption,weight, and dimensions as fundamental to favor the portabilityin field applications.
II. BASIC THEORY OF MICROWAVE QSRS
In a geometrically perfect spherical resonator with a perfectlyconducting surface, the electric and magnetic vector fields Φtake the form [17]
Φlnm(r, θ, φ) = Jl(ξσlnr/a)Ylm(θ, φ) (1)
where a is the cavity radius, Jl is a spherical Bessel functionof order l, Ylm is a vector spherical harmonic, ξσln is aneigenvalue obtained with arbitrary precision by imposing theappropriate boundary condition for TM or TE modes (speci-fied by the superscript σ), and the set of subscripts (l, n,m)uniquely defines the mode eigenfunction, with the selectionrules (l, n) = 0, 1, . . ., and m = 0, 1, . . . , l. The correspondingeigenfrequencies fσ
ln are at least triply degenerate for the casel = 1 and m = 0, ±1 and given by
fσln =
cξσln2πa
(2)
where c is the speed of light in the medium which fills thecavity. The practical realization of a spherical resonator and theultimate precision and accuracy achievable in the experimentaldetermination of its eigenfrequencies necessarily deals with thedegeneracy implied by (1) and (2), and the quest for fabricationof geometrically “perfect” spheres inevitably fails due to thecurrent limits of machining procedures of metallic artifacts.This difficulty was successfully overcome by the demonstrationthat the intentional lift of the degeneracy by using a deformedgeometry, typically transforming the sphere into a triaxial ellip-soid with slightly different axes, separates the resonances of themodes with l = 1 into three weakly overlapping singlets, whichcan be more easily measured [5]. If the shape of the cavity isdefined by the equation
x2
a2(1 + ε2)2+
y2
a2+
z2
a2(1 + ε1)2= 1 (3)
with 0 < (ε1, ε2) � 1, the resulting fractional amount of thesplitting can be analytically calculated by theory at first andsecond orders in the scale parameter ε [4], [18]. As an example
TABLE IEIGENVALUES OF A PERFECT SPHERE AND A TRIAXIAL ELLIPSOID
of the effect of such geometrical perturbation, the eigenvaluesof modes TM11 to TM13 and TE11 to TE13 are reported inTable I for a perfect sphere and a triaxial ellipsoid defined by(3) with ε1 = 2ε2 = 2 · 10−3. In addition to geometrical per-turbations, the finite conductivity of the internal cavity surfaceand the consequent increase of the experimentally determinedfrequencies by skin effect, as well as the perturbing effect ofantennas and ducts, must be kept into account for accuratework with QSRs. Accurate models of all these effects have beenrecently worked out [19].
III. REQUIREMENTS OF QSRFREQUENCY MEASUREMENTS
In spite of the slight frequency perturbation induced by thequasi-spherical geometry, the relative separation of the singlecomponents of the triplets (200 ppm in the example consid-ered earlier), combined with the high quality factors of theresonances (between 6 · 104 for mode TM11 and 1.6 · 105 formode TE13 in a copper cavity with a radius of 5 cm), allowsone to achieve precision at a few ppb level if the signal/noiseratio in the measurement at a single discrete frequency andthe performance of the fitting function and algorithm (seeSection V) over a band of a few megahertz can be maintainedon the order of 1 · 103. However, while these conditions can bereadily met by using VNAs, they represent a very challengingtarget for an LC instrument.
Measurements of the frequency and half-width of the res-onances are typically performed acquiring the transmissionscattering coefficient S21 of the cavity at a discrete set offrequency values (e.g., 201 points) over a band approximatelycentered at the observed mean frequency of the triplet. Themain issues involved with this task are as follows.
1) Working band and resolution: The internal radius of aQSR in reported applications typically varies between 3and 9 cm, with the lowest order TM and TE resonancesoccurring over the wideband between 1.5 and 7 GHz.Such frequency range translates the aimed relative ac-curacy into an absolute frequency accuracy in the range50–300 Hz.
2) Dynamic range: The amplitude difference between peaksand floor within the triplet is usually in the range of30–40 dB. In addition, different levels of transmissionthrough the cavity can be encountered: In many cases, thetransmission is intentionally kept quite low in order not tooverload the cavity. Therefore, considering an additionalrange of 20 dB which keeps into account the variabilityof transmission levels for different resonators, an overalldynamic range of 60 dB is required.
3) Type of measurement: Despite that scalar measurementsmay be enough to determine the resonance frequency
CORBELLINI AND GAVIOSO: LOW-COST INSTRUMENT FOR THE ACCURATE MEASUREMENT OF RESONANCES 1261
Fig. 1. Block diagram of the proposed system. The architecture consists ofthree main elements: The processing unit, the signal generator, and the receiver.
in many cases, vectorial measurements are necessary tomeet high accuracies due to the necessity to compensatefor systematic errors.
4) Computation capability: In order to assure the portabilityof the instrument, which is necessary for field applica-tions, a suitable control unit has to be included so thatthe instrument can autonomously execute measurement,process the gathered data, and display the results.
IV. PROPOSED ARCHITECTURE
With respect to conventional VNAs, the architecture of theproposed instrument was greatly simplified, avoiding all thosecomponents that are not necessary for the measurement ofthe resonances (Fig. 1). In particular, components related tothe measurement of the three scattering coefficients [20], [21]S11, S12, and S22 are not strictly necessary. A further sim-plification was obtained by considering that the transmissioncoefficient S21 can be measured by using an arbitrary nor-malization factor (i.e., resonances can be also obtained fromthe data set K · S21(f), where K is an unknown complexconstant). However, the use of a simplified architecture andLC components usually leads to poor performance in terms ofband/resolution ratio, dynamic range, and phase noise. In theproposed system, these problems are tackled by using three keysolutions:
1) a microwave signal generator based on a special referencefrequency generator, which permits one to increase theresolution of LC wideband microwave synthesizers, asexplained in the following section;
2) a simple single downconversion receiver equipped with awide-gain-range programmable amplifier (PGA), whichexpands the dynamic range;
3) a digital processing unit based on a DSP that implementsa coherent subsampling and a sin-fit algorithm, whichtogether can provide vectorial measurements with greatimprovement of the signal/noise ratio at LC.
A. Signal Generator
Different low-priced microwave synthesizers are available onthe market, but those with wide working band usually exhibitresolutions not better than 1 MHz, which is not adequate for the
Fig. 2. Block diagram of the receiver. A single downconversion is employedto convert the microwave signals to the IF.
proposed application. Such synthesizers multiply a referenceclock, typically of 10 MHz, by a limited set of factors througha phase-locked-loop device to synthesize the microwave signal.To overcome the problem in the proposed system, we decided toincrease the resolution of the generator by finely changing thereference clock instead of using a device with more availablefactors. As an example, a resolution of 100 Hz can be achievedfrom a synthesizer with a 1-MHz step by changing its referenceclock by 1 Hz from 10 to 10.000001 MHz when the synthesizeris programmed to multiply by 100.
Fortunately, the realization of such finely tunable clocks isnowadays made possible, economically, by means of directdigital synthesizers (DDSs). We selected the Analog Device’sDDS AD9954 to realize a generator whose output can finelychange in the range 10 MHz ± 50 kHz with a step of 0.1 Hz.This setup allows one to achieve a final resolution of 70 Hzover the whole desired band of 1–7 GHz. Three LC synthesizersby Synergy Microwave Corporation were selected to cover thewhole band, namely, LFSW80120-100, LFSW190410-100, andLFSW397697-100; each of them spans over about one octaveand exhibits a frequency step of 1 MHz. The system is able toautomatically select, activate, and switch the correct synthesizeraccording to the working frequency.
Two identical signal generators were realized to generateboth the stimulus for the cavity (RF generator) and the demod-ulating signal for the receiver [local oscillator (LO) generator].
B. Receiver
The architecture of the receiver is shown in the block di-agram in Fig. 2. A single downconversion is employed toconvert the microwave signals (reference and input signals) toan intermediate frequency (IF) of 1 MHz that can be directlyacquired by the processing unit. To improve isolation betweenports, and therefore not impairing the dynamic range of thereceiver, a power splitter with good isolation and a sequence ofamplifier–attenuator are used to feed the LO ports of the mixers.In addition, to further extend the dynamic range of the receiver,
1262 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 62, NO. 5, MAY 2013
Fig. 3. Photograph of the prototype during a measurement with a QSR. Theresonance is shown in real time on the graphical LCD.
a PGA with a wide range of gains (more than 70 dB) is insertedafter the mixer to amplify the IF signals. The receiver is realizedby using LC parts provided by Mini Circuits, and the amplifiersare realized by using the Analog Devices AD8369 PGA.
C. Processing Unit
The processing unit is based on the 32-b CPUAT32UC3C0512C by Atmel. This device can run up to 66 MHz,embeds a 32-b floating unit, and exhibits all the features neces-sary to implement the required functions. In particular, it em-beds a differential 2-MSa/s A/D converter along with a doublesample and hold that permits a simultaneous subsampling of thetwo IF signals. For the realization of the instrument prototype,the evaluation board AT32UC3C-EK has been employed. Theboard, which is visible in Fig. 3, is equipped with a graphicalliquid crystal display (LCD), different I/O devices, and 256 MBof synchronous dynamic random access memory. The DSP hasbeen programmed to control all the devices of the system, toperform the coherent acquisition, and to process the acquireddata. This system is able to achieve a measurement speed ofabout 400 μs per frequency point per average cycle (e.g., 201frequency points with 32 averages require less than 3 s). Theinstrument can work as a stand-alone device, showing theresults on the graphical LCD and saving the data on a microSDmemory card. Alternatively, it can be interfaced with a personalcomputer. For this latter case, a specific computer applicationhas been realized to control the instrument and to receivethe measurements in real time. The unit embeds an internalreference clock that is used to synchronize the acquisitionand the microwave synthesizers. However, for the mostdemandingly accurate measurements, an external frequencystandard can be connected to synchronize the instrument.
V. DATA FITTING
In order to be able to achieve accuracies of a few ppb in thedetermination of the resonance frequencies, the strategy of sim-ply locating the maxima in the transmission coefficients S21(f)
TABLE IIPERFORMANCE OBTAINED WITH DIFFERENT BACKGROUND MODELS
is not adequate, even if the resolution of the instrument iscomparable with the target accuracy, because of the following:
1) Precision and accuracy in the identification of the max-imum would be directly limited by the amount of noisewhich affects the signal not fully exploiting the advantagerepresented by the high quality factor of the resonances.
2) The maximum of a single peak within the triplet signifi-cantly differs from its mean frequency, which is the quan-tity of interest in most accurate applications of QSRs.
To overcome these problems, the triple resonance is usuallyacquired with lower resolution (kilohertz range) but over alarger band, and successively, the measurements are fitted byusing a suitable model, corresponding to the sum of threecomplex Lorentzian-like functions, first proposed for accuratefitting of acoustic resonances in [22] and later adapted formicrowave resonances in [23]
S21(f) =3∑
i=1
Aif
f2 − (F i)2+
2∑i=0
Bi · (f − f ∗)i (4)
where f ∗ represents the central frequency of the measurementband, F i represents the complex resonance frequencies to beidentified, and Bi represents the coefficients that describe thebackground contribution of the acquired S21 parameter. Thebackground function accounts for the overlapping contributionfrom the tails of modes other than the one of interest, as theseare inevitably present in a system with multiple resonances.Thus, the order of the needed background expansion dependson the spanned frequency range. This model, which is wellphysically founded, is capable of providing extremely accuratedeterminations of the resonance parameters for data acquiredwith a calibrated conventional VNA (in most cases, calibrationis not even needed if the frequency band required to span asingle mode is limited to a few megahertz).
Unfortunately, the instrument proposed in this work cannotbe fully calibrated since its architecture does not allow all thescattering parameters to be measured. In addition, the simplereceiver architecture limits the isolation between the instrumentports to only 70–80 dB leading to nonnegligible errors whenmeasuring weakly coupled resonances. In reason of these lim-its, even when a quadratic background is sufficient to accountfor nonlinearities in the data acquired with the VNA, for theanalysis of the data acquired with the instrument proposedhere, it was found to be necessary to extend the backgroundexpansion up to cubic order, particularly for compensating theeffects of the limited isolation. The effect of the alternativechoice of different orders in the background function is furtherdiscussed in Section VI and listed in Table II.
A minimization Lavenberg–Marquardt algorithm, developedin MATLAB software, implemented the linearized regression
CORBELLINI AND GAVIOSO: LOW-COST INSTRUMENT FOR THE ACCURATE MEASUREMENT OF RESONANCES 1263
procedure needed for the determination of the 20 fitted parame-ters in the modified equation (4) and their statistical uncertain-ties. The algorithm minimizes the sum of the squared residualsseparately calculated for the real and imaginary parts [22]. Ina typical scan of a QSR triplet with a VNA (201 frequenciesspanning 10 MHz; IFBW = 10 Hz), the fractional uncertaintyof the fitted value of the real part of F i, i.e., the mean resonancefrequency 〈f〉 of the triplet, is easily maintained below 20 ppb.
VI. EXPERIMENTAL RESULTS
A first prototype of the instrument was realized and tested bymeasuring the resonance frequencies of the modes TM11 andTE12 of a 6-cm-radius copper QSR maintained at ambient con-ditions of temperature and pressure. Fig. 3 shows the arrangedbench. Some relevant results achieved during these initial tests,including a comparison of the S21 measurements between theprototype and a laboratory VNA, were discussed in [24].
To evaluate the accuracy achievable by the designed instru-ment in the determination of the resonance parameters, weinitially compared its performance in the acquisition of modeTM11 for the same cavity used in [24]. The acquired data andthe results of their analysis using (4) with cubic backgroundwere compared with those obtained using a laboratory VNA(HP8510C). In order to avoid problems related to the accuracyof the clocks employed by two instruments, both the VNAand the designed instrument were synchronized using a shared10-MHz external source.
The test consisted in a number of successive acquisitionsof TM11 mode data, while the cavity was filled with air andallowed to drift with the variation of ambient temperature.In these conditions, the highest peak in the triplet falls atabout 2.181 GHz with an amplitude of −30 dB. The datawere repeatedly recorded as a function of time, first withthe designed instrument and successively with the referenceVNA (HP8510C), over an interval of 30 min. Assuming thatthe effects of small variations of ambient pressure and aircomposition are negligible over such limited period of time, therecorded negative drift of mode TM11 (Fig. 4) is determined bytemperature variations as the result of two competing effects:the thermal expansion of the cavity in response to an increasein ambient temperature and the decrease of the permittivitydue to the decrease of the gas density. For air in a copperresonator, the former effect predominates, as the thermal ex-pansion coefficient (1/a)(∂a/∂T ) ∼= 17 · 10−6 K−1 of copperis much larger than the sensitivity of the permittivity of airto temperature variations (1/εair)(∂εair/∂T ) ∼= 2 · 10−6 K−1.As a consequence of the high thermal inertia of the resonator,the negative drift of the frequency data recorded from bothinstruments is slow and well approximated by a linear functionof time. The fitted slope of this linear trend allows one toestimate the rate of temperature change at about 0.4 K · h−1,while the visual inspection of the residuals from this fit (Fig. 5)provides a good check of the agreement between the VNAand our prototype. Remarkably, such comparison shows noevidence of systematics, and the scatter of the residuals for theprototype is less than 30 ppb.
Fig. 4. Drift of several consecutive measurements of the resonance frequencyof mode TM11 alternatively observed with the proposed instrument and with aclassical VNA. The apparent frequency drift is due to an increase of ambienttemperature within the laboratory room and is an approximately linear functionof time. The temperature change reported on the (right) vertical axis exemplifiesthe device sensitivity.
Fig. 5. Fractional deviations of the experimental frequencies of mode TM11
corrected for the temperature drift shown in Fig. 4. The standard deviation ofthe set of measurements obtained with the designed instrument is 15 ppb.
Following this initial test, to minimize resonance drifts andenhance the meaningfulness of the comparison, we arrangedan experimental setup capable of providing more stable condi-tions. The setup comprised a gold-plated stainless steel triaxialellipsoid resonator1 (Fig. 6) with an average radius of 2.54 cmand ε1 = 1 · 10−3 and ε2 = 3 · 10−3 [9].
The cavity was inserted in a steel vessel and kept under amoderate vacuum (< 1 · 10−2 mbar) while being thermostattedat (9 ◦C ± 0.5 ◦C) in a small portable refrigerating box unit.Although the imperfections of this rough arrangement did not
1This cavity was designed and realized at the National Institute for Standardsand Technology (NIST) and was loaned to Istituto Nazionale di RicercaMetrologica (INRiM) by courtesy of the Fluid Science Group of NIST.
1264 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 62, NO. 5, MAY 2013
Fig. 6. QSR used as a test bench for comparison of the performances of theVNA and the designed instrument. (Above) Two hemispheres comprising thestainless steel microwave cavity. Gold plating enhances the resonance qualityfactors and provides optimum resistance to corrosion in microwave hygrometryexperiments and applications. (Below) Assembled resonator is inserted withina vacuum- and pressure-tight vessel for measuring the dielectric constant as afunction of temperature and pressure.
completely eliminate the drift of the resonance frequencies,which were still observable as oscillating fluctuations around amean value at a rate of about 200 ppb · h−1, the improvement ofthe frequency stability allowed a more direct and reliable com-parison between the designed instrument and a VNA (AgilentE5071C). Both the instruments were synchronized by using anexternal rubidium frequency standard (SRS FS725).
The mode TM11 at 5.15 GHz was selected to perform theanalysis. The recorded S21 data for a single acquisition of thismode alternatively obtained using the designed instrument andthe VNA are compared in Fig. 7: In this case, the maximumamplitude of the resonance is only −47 dB, and most of the ac-quired signal lies below the 70–80-dB level, which is compara-ble with the isolation between the instrument ports; as expected,the two instruments provide quite different measurements ofS21 as soon as the amplitude approaches and goes below thislimit. However, we expected that this effect would have beenaccounted for and corrected by the background terms in thefitting model of (4). This is confirmed by the fit residuals shownin Fig. 8: The standard deviation of the residuals is only 0.05 dBfor the amplitude and 0.35◦ for the phase. To further highlightthe benefit of the cubic background model, the residuals werecomputed also by using background functions of lower orderas summarized in Table II. It is worth to note that, due to thelevel of noise, no significant improvements are observed on the
Fig. 7. Transmission coefficients S21 of mode TM11 in a QSR measured withthe proposed instrument and a conventional VNA. A portion of the acquiredsignal falls below the −80-dB value, which is comparable with the achievedlevel of isolation between ports.
Fig. 8. Amplitude and phase residuals from a fit to TM11 mode data recordedusing the designed instrument. The standard deviation of the residuals is0.05 dB for the amplitude and 0.35◦ for the phase.
amplitude of the residuals moving from a quadratic to a cubicbackground model; however, the discrepancy between the fittedresonance frequencies with the two instruments is reduced from120 to 40 ppb. No further improvements were observed furtherincreasing the order of the background function. Fig. 9 shows acomparison of the TM11 frequency fitted from data recordedwith the designed instrument and the VNA for consecutivemeasurements over a one-hour time lapse.
In spite of the imperfections of the thermostatting systemwhich are apparent from the slight temporal drift of the ob-served frequencies, the quality of the comparison is indeedremarkable. The relative differences are well below the 50-ppblevel, which was taken as the initial required specification.
In a final experiment, a comparison of our device and a VNA(Agilent E5071C) was based on a determination of the permit-tivity of helium. For this test, we used an apparatus developed
CORBELLINI AND GAVIOSO: LOW-COST INSTRUMENT FOR THE ACCURATE MEASUREMENT OF RESONANCES 1265
Fig. 9. Mean resonance frequency of the TM11 mode obtained from fittingS21 data with (4) and a cubic background. Some thermal effects are still visiblein the results. The relative difference between the mean values, alternativelyobtained with the designed instrument and the VNA, is lower than 50 ppb.
at INRiM to pursue a determination of the Boltzmann constantk. The features of this experiment are well documented [19],[25], and they provide a stable and accurate reference for thedensity of a gaseous sample within a copper QSR.2 Initially,we alternatively used the VNA and our LC device to recorddata for the TM12 resonance excited within the cavity while thiswas being flushed at 120 sccm with a sample of He maintainednear 100 kPa and 273.16 K by suitable controls. The rawexperimental S21 data were fitted to the model in (4) to providerepeated estimates of the mean frequency 〈f〉 and half-width〈g〉 of the TM12 triplet, whose sum 〈f + g〉100 kPa accountsfor the skin effect induced by the finite resistivity of copper.Successively, the cavity was thoroughly evacuated, and after atime lapse of several hours needed to recover the temperaturevariation induced by adiabatic cooling, a similar data record ofthe quantity 〈f + g〉0 was collected near 273.16 K. The plotsin Fig. 10 show the fractional deviations of several repeatedacquisitions of these quantities with the VNA and the LC fromthe same reference value (see the caption in Fig. 10) at 100 kPaand in vacuum. In this experiment, the comparison betweenthe VNA and the LC is very challenging since the resonatorantennas are intentionally kept extremely weakly coupled. Thetransmission coefficients at the resonance peaks are −75, −83,and −85 dB, while the floor between peaks reaches −100 dB.Remarkably, no systematic difference is apparent between theresults obtained from the two instruments.3 To quantify thequality of the agreement, we note that the fractional differencebetween the mean of 16 repeated acquisitions of 〈f + g〉VNA
and a similar record of 〈f + g〉LC is only −23 ppb at 100 kPaand 44 ppb in vacuum, making, in both cases, these quanti-ties consistent within one standard deviation. All these data
2This cavity was designed and realized at Laboratoire National de métrologieet d’Essais and was loaned to INRiM by courtesy of the Thermometry Division.
3The LC instrument was configured to employ an averaging factor of 400,corresponding to an integration time of about 150 ms and to an overall measure-ment time of about 30 s, which was comparable with the VNA measurementtime.
Fig. 10. Fractional deviations of the mean corrected frequencies of the TM12
mode recorded at 100 kPa and in vacuum, at 273.16 K in a copper QSR [25]with a VNA and the instrument described in this work. (Solid symbols) At100 kPa, (black line) the reference value equals 5806.346362 MHz; in (hollowsymbols) vacuum, the reference value equals 5806.543692 MHz.
may be combined to provide an experimental estimate of thepermittivity of helium εHe at 100 kPa and 273.16 K exactly,upon applying corrections to compensate for the differencesbetween the experimental temperature Texp and 273.16 K, fromthe relation
εHe =1
μHe
(〈f + g〉0
〈f + g〉100 kPa · (1 +KT /3)
)2
(5)
where the relative magnetic permeability of helium μHe at100 kPa may be neglected and a relevant correction depends onKT , the isothermal compressibility of copper [5]. Finally, using(5) for VNA and LC data, we obtain the consistent estimates:106 × (εVNA − 1) = 68.334± 0.104 and 106 × (εLC − 1) =68.218± 0.145, which well agree with the expected value106 × (εtheory − 1) = 68.289. The latter can be obtained withnegligible uncertainty from published ab initio calculations ofthe thermodynamic and electromagnetic properties of He (see[6] and references therein).
VII. CONCLUSION
Quasi-spherical microwave resonators may be usefully em-ployed in many industrial metrological applications, if theconvenience of the determination of their resonance frequenciesmay be increased. Working toward this goal, an LC solutionfor the realization of an accurate instrument for measuring theresonance frequency of microwave cavity resonators has beenpresented.
We remark that the demonstrated performance of the pro-posed solution, already at a prototype level, indicates that itwould satisfy the accuracy requirements of industrial applica-tions as well as those of many demanding scientific applica-tions. The overall cost of the realized prototype is less than$2000, and it can be even decreased if the final applicationwould not require the large frequency band and the dedicateduser interface.
1266 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 62, NO. 5, MAY 2013
Further scheduled improvements of the performance have amajor goal in increasing the operating frequency range, nowlimited to 7 GHz, while maintaining the cost at a comparablelevel. Finally, we remark that, although, in this work, theattention has been focused on the possible application andthe specific requirements of fluid metrology with microwavecavities, the designed instrument has a potential interest formany other microwave applications.
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Simone Corbellini was born in Italy in 1977. He received the M.S. degree inelectronic engineering and the Ph.D. degree in metrology from Politecnico diTorino, Torino, Italy, in 2002 and 2006, respectively.
He is currently with the Dipartimento di Elettronica e Telecomunicazioni,Politecnico di Torino. His main fields of interest are digital signal processing,distributed measurement systems, and intelligent microcontroller-based instru-mentation. He is currently working on the development of instrumentation forthe measurement of water content in energy gases.
Roberto M. Gavioso was born in Italy in 1968. He received the M.S. degree inphysics from the Università degli Studi di Torino, Torino, Italy, in 1993 and thePh.D. degree in metrology from Politecnico di Torino, Torino, in 1997.
Since 1998, he has been a Research Scientist with the ThermodynamicsDivision, Istituto Nazionale di Ricerca Metrologica, Torino. He leads a researchgroup mainly active in the field of physical acoustics. His recently publishedwork and current main interests deal with metrological applications of acousticand microwave resonators, including the development of primary standards oftemperature, pressure, and humidity.
